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The product of two consecutive even natural numbers is equal to $2024$ . What is the sum of these integers?

Hint

Let the even numbers be

2 n and 2 n + 2

You need to solve

2 n (2 n + 2) = 2024

Hint

Recall that natural numbers are positive.

Solution

Let the even numbers be

2 n and 2 n + 2

then we need to solve

\begin{aligned} 2 n (2 n + 2) & = 2024 \\ 4 n (n + 1) & = 2024 \\ n (n + 1) & = 506 \\ n^{2} + n - 506 & = 0 \end{aligned}

Whilst this can be factorised, it's more convenient to use the formula as $506$ is a large number.

\begin{aligned} n & = \frac{- 1 \pm \sqrt{1 - 4 (1) (- 506)}}{2} \\ = \frac{- 1 \pm 45}{2} \\ n & \in {- 23, 22} \end{aligned}

However $n \in N$ so $n = 22$ .

The sum of the even numbers is $44 + 46 = 90$ .